数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 1817-1830.doi: 10.1007/s10473-022-0507-z

• 论文 • 上一篇    

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

Narimane AISSAOUI1, Wei LONG2   

  1. 1. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China;
    2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China
  • 收稿日期:2021-03-12 修回日期:2022-05-19 发布日期:2022-11-02
  • 通讯作者: Wei Long,E-mail:lwhope@jxnu.edu.cn E-mail:lwhope@jxnu.edu.cn
  • 基金资助:
    The second author was supported by NSF of China (11871253), supported by Jiangxi Provincial Natural Science Foundation (20212ACB201003), Jiangxi Two Thousand Talents Program (jxsq2019101001), Double-high talents in Jiangxi Province and Jiangxi Provincial Department of Education Fund (GJJ191687).

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

Narimane AISSAOUI1, Wei LONG2   

  1. 1. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China;
    2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China
  • Received:2021-03-12 Revised:2022-05-19 Published:2022-11-02
  • Contact: Wei Long,E-mail:lwhope@jxnu.edu.cn E-mail:lwhope@jxnu.edu.cn
  • Supported by:
    The second author was supported by NSF of China (11871253), supported by Jiangxi Provincial Natural Science Foundation (20212ACB201003), Jiangxi Two Thousand Talents Program (jxsq2019101001), Double-high talents in Jiangxi Province and Jiangxi Provincial Department of Education Fund (GJJ191687).

摘要: This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms: $$ \left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$$ Here $a,b$ are positive constants, $V(x),Q(x)$ are positive radial potentials, 1<p<5, $\varepsilon$ >0 is a small parameter, $f(x)$ is an external source term in $L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$.

关键词: nonlinear Kirchhoff problem, Lyapunov-Schmidt reduction, positive solutions, level set

Abstract: This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms: $$ \left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$$ Here $a,b$ are positive constants, $V(x),Q(x)$ are positive radial potentials, 1<p<5, $\varepsilon$ >0 is a small parameter, $f(x)$ is an external source term in $L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$.

Key words: nonlinear Kirchhoff problem, Lyapunov-Schmidt reduction, positive solutions, level set

中图分类号: 

  • 35J20